<?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>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:9"/><i><font color="Green">E10</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="rusub_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:9"/><i><font color="Green">E11</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</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="E3:9_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:9_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:9_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E6:9_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E5:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:9_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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="E9:9_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E10:9_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E11:9_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E12:9_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E9:9_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E13:9_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_2.html#E4:9_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rusub_2.html#E10:9_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E14:9_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E15:9_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:9_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_2.html#E3:9_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rusub_2.html#E11:9_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rusub_2.html#E12:9_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</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="E3:9_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:9_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:9_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E6:9_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E4:9_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:9_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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="E9:9_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E10:9_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E11:9_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E12:9_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <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#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E9:9_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E13:9_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_2.html#E5:9_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="rusub_2.html#E11:9_2"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E14:9_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E15:9_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:9_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_2.html#E3:9_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rusub_2.html#E10:9_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rusub_2.html#E12:9_2"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:9"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_2.html#E2:9"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:9"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E1:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rusub_1.html#T24">RUSUB_1:24</a></i>;<br/>


</div>
