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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V7">RealLinearSpace-like</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">M</font> be  <a href="rusub_4.html#V2" target="_self">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">N</font> be  <a href="rusub_4.html#V2" target="_self">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:55"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><br/> for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K7" target="_self">+</a> <font color="Maroon">N</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K7" target="_self">+</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</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> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K7" target="_self">+</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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 <font color="Maroon">V</font>, <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:55_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K7" target="_self">+</a> <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K7" target="_self">+</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E2:55_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> ) </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> ) </span>}</span>  )
 
;<br/>
<b>then consider </b><font color="Maroon">u1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:55_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 ;<br/>
<b>consider </b><font color="Maroon">u2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:55_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="txt" href="rusub_4.html#E4:55_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E7:55_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</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">u2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="txt" href="rusub_4.html#E6:55_1"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<a NAME="E8:55_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</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">v2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="txt" href="rusub_4.html#E6:55_1"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</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">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">u2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="rusub_4.html#E6:55_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></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">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span></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">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E10:55_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</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">y</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_4.html#E7:55_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rusub_4.html#E8:55_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:55_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</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">y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="rusub_4.html#K7" target="_self">+</a> <font color="Maroon">N</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:55"/>
<font color="Maroon">M</font> <a href="rusub_4.html#K7" target="_self">+</a> <font color="Maroon">N</font> is <a href="rusub_4.html#V2" target="_self">Affine</a>
 <b>by </b><i/>;<br/>


</div>
