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

<b>let </b><font color="Maroon" title="c1">V</font> be  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a> <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/>





<b>assume </b><a NAME="E1:44"/>
<font color="Maroon" title="c1">V</font> <a href="rlsub_2.html#R1" title="RLSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c2">W1</font>,<font color="Maroon" title="c3">W2</font>
 ;<br/>

<a NAME="E2:44"/><b>then </b><i><font color="Green" title="E35">A1</font></i>: 
(  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U4</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> )
 
<b>by </b><i><a class="ref" href="rlsub_2.html#D4" title="RLSUB_2:def.4">RLSUB_2:def 4</a></i>;<br/>
<b>then </b> <a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span> = 
 <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
<b>by </b><i><a class="ref" href="rlsub_1.html#D4" title="RLSUB_1:def.4">RLSUB_1:def 4</a></i>
<br/>.= 
 <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_1.html#T48" title="RLSUB_1:th.48">RLSUB_1:48</a></i>
;<br/>
<a NAME="E4:44"/><b>then </b>
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rlvect_5.html#T33" target="_self" title="RLVECT_5:th.33">Th33</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c3">W2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="rlvect_5.html#T36" target="_self" title="RLVECT_5:th.36">Th36</a></i>
<br/>.= 
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K2" title="RLSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_5.html#E2:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="rlsub_1.html#D4" title="RLSUB_1:def.4">RLSUB_1:def 4</a></i>
<br/>.= 
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c1">V</font>
<b>by </b><i><a class="ref" href="rlvect_5.html#T31" target="_self" title="RLVECT_5:th.31">Th31</a></i>
;<br/>
<b>hence </b><a NAME="E6:44"/>
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 ;<br/>


</div>
