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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E6:19"/>
 <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is    <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>
<a NAME="E1:19_1"/><i><font color="Green">E21</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E2:19_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rusub_1.html#E3:19"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E3:19_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rusub_1.html#E4:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E4:19_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<a NAME="E1:19_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rusub_1.html#E5:19"><i><font color="Green">E20</font></i></a>, <a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1"/>
(  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="bhsp_1.html#V2">RealUnitarySpace-like</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V7">RealLinearSpace-like</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V6">right_complementable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:19_1_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font>

<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:19_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <font color="Olive">b<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="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>14</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>12</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="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)</span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:19_1_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)ex <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:19_1_2_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1_2_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) ;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rusub_1.html#E3:19_1_2_4"><i><font color="Green">E29</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>



<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:19_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>



<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:19_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>



<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>

<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:19_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) holds 1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>

<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:19_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)<br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) implies <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:19_1_2_9"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_2_9_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:19_1_2_9_1"/><b>then </b>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E3:19_1_2_9_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<a NAME="E4:19_1_2_9_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_2_9_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)
 <b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1_2_9"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) implies <font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_2_9_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)
 ;<br/>

<a NAME="E2:19_1_2_9_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E3:19_1_2_9_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E4:19_1_2_9_2"/><b>then </b>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_2_9_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1_2_9"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) )
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1_2_9"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:19_1_2_9"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_9_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E2:19_1_2_9_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_2_9_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E8:19_1_2_9"/>
<font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_4"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_9_4"/>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<a NAME="E2:19_1_2_9_4"/><b>then </b>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<a NAME="E3:19_1_2_9_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_2_9_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E9:19_1_2_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_5"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_9_5"/><i><font color="Green">E34</font></i>: 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<i><font color="Green">E35</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font> = 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E1:19_1_2_9_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:19_1_2_9_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_1.html#E2:19_1_2_9_5"><i><font color="Green">E35</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E10:19_1_2_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_9_6"><b>proof </b></a><div class="add">

<i><font color="Green">E34</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font> = 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rusub_1.html#E3:19_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>
;<br/>
<font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
<font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rusub_1.html#E4:19_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:19_1_2_9_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:19_1_2_9_6"><i><font color="Green">E34</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:19_1_2"/>
(  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="bhsp_1.html#V2">RealUnitarySpace-like</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V7">RealLinearSpace-like</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="rlvect_1.html#V6">right_complementable</a> )
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:19_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rusub_1.html#E2:19_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rusub_1.html#E3:19_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rusub_1.html#E4:19_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="rusub_1.html#E5:19_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rusub_1.html#E6:19_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rusub_1.html#E7:19_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rusub_1.html#E8:19_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a>, <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1"/>
 <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is    <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rusub_1.html#E3:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rusub_1.html#E4:19"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rusub_1.html#E5:19"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rusub_1.html#E1:19_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rusub_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:19"/>
 <a href="bhsp_1.html#G1">UNITSTR</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is    <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
