<?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">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W1</font> be  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W2</font> be  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/>





<b>thus </b><a NAME="E1:72"/>
( <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font> implies <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:72_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:72_1_1"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:72_1_1"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:72_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:72_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font>;<br/>
<a NAME="E3:72_1_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:72_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <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="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> </span>}</span> 
 
;<br/>
<a NAME="E5:72_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E6:72_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <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="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:72_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E9:72_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 ;<br/>
<a NAME="E10:72_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E11:72_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E4:72_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T19" target="_self">Th19</a></i>;<br/>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font>;<br/>
<a NAME="E6:72_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:72_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <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="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> </span>}</span> 
 
;<br/>
<a NAME="E8:72_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E9:72_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <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="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:72_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E12:72_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 ;<br/>
<a NAME="E13:72_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E14:72_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:72_1"/>
<font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:72"/>
( <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> implies <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font> )
 ;<br/>


</div>
