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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>



<i><font color="Green" title="E6">A1</font></i>:  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_4.html#D6" title="RUSUB_4:def.6">RUSUB_4:def 6</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span> 
<b>by </b><i><a class="ref" href="rlsub_2.html#D1" title="RLSUB_2:def.1">RLSUB_2:def 1</a></i>
;<br/>
<a NAME="E2:15"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">u</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:15_2"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">u</font> &amp; <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> )
 <b>by </b><i><a class="txt" href="convex2.html#E1:15"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<a NAME="E4:15_2"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="txt" href="convex2.html#E3:15_2"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:15_2"/><b>then </b>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="ref" href="rusub_4.html#D6" title="RUSUB_4:def.6">RUSUB_4:def 6</a></i>;<br/>
<a NAME="E6:15_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">v'</font></span>)</span> where <font color="Olive" title="b1">u'</font>, <font color="Olive" title="b2">v'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="convex2.html#E3:15_2"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:15_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
 <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:15"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 ;<br/>

<a NAME="E2:15_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:15_3"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">v</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font> )
 ;<br/>
<a NAME="E5:15_3"/>
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="txt" href="convex2.html#E4:15_3"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="rusub_4.html#D6" title="RUSUB_4:def.6">RUSUB_4:def 6</a></i>;<br/>
<a NAME="E6:15_3"/><b>then </b>
( <font color="Maroon" title="c5">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:15_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E1:15"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="convex2.html#E4:15_3"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:15"/><b>then </b>
<span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:15"/>
 <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E3:15"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
