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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="rusub_2.html#D6" target="_self" title="RUSUB_2:def.6">Def6</a></i>;<br/>
<a NAME="E3:59"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c1">V</font> <a href="rusub_2.html#R1" title="RUSUB_2:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E1:59"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="rusub_2.html#E53"><i><font color="Green" title="E36">Lm15</font></i></a></i>;<br/>
<a NAME="E4:59"/><b>then </b><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_2.html#D6" target="_self" title="RUSUB_2:def.6">Def6</a></i>;<br/>
<a NAME="E5:59"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> )
 
<b>by </b><i><a class="txt" href="rusub_2.html#E3:59"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="rusub_2.html#D6" target="_self" title="RUSUB_2:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:59"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rusub_2.html#K4" title="RUSUB_2:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <b>by </b><i><a class="txt" href="rusub_2.html#E1:59"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="rusub_2.html#E2:59"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="rusub_2.html#E4:59"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="rusub_2.html#T45" target="_self" title="RUSUB_2:th.45">Th45</a></i>;<br/>


</div>
