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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">V1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">D</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">d1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">M</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">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">D</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font>
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</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">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>set </b><font color="Maroon">W</font> =  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>
<a NAME="E5:26"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:26"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</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/>
<a NAME="E7:26"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">x</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/>
<a NAME="E8:26"/>
(  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V6">right_complementable</a> &amp;  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V7">RealLinearSpace-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">AV</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>;<br/>
<b>set </b><font color="Maroon">MV</font> = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>;<br/>
<b>thus </b><a NAME="E1:26_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #), <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E2:26_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #), <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #), <font color="Maroon">z</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font>, <font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <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">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</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">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z1</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z1</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">V</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E3:26_1"/>
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font>, <font color="Maroon">z</font> =  <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #)</span>)</span> = 
<font color="Maroon">y</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">V</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>


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

<b>thus </b><a NAME="E4:26_1"/>
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #)  ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) st <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:26_1_4"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<font color="Maroon">v</font> = 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">x1</font>
<b>by </b><i>, <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">x1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</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">x</font>
<b>by </b><i/>
;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E6:26_1_4"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #)
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E5:26_1"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">y</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #), <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>



<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> = 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</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">V</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">V</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x1</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</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">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E6:26_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K3">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>





<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <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">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</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">V</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">V</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</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">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E7:26_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>





<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <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">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</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">V</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>


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

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, <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">x</font> = 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</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>
<b>hence </b><a NAME="E9:26"/>
 <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) is    <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>
 <b>by </b><i>, , , , </i>;<br/>


</div>
