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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <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="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font> holds <br/><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="Olive" title="b2">W2</font>,<font color="Olive" title="b1">W1</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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="c3">W2</font>,<font color="Maroon" title="c2">W1</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c3">W2</font>,<font color="Maroon" title="c2">W1</font></span><br/>

<a NAME="E2:43"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D4" target="_self" title="RLSUB_2:def.4">Def4</a></span>;<br/>
<a NAME="E3:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c3">W2</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c2">W1</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T14" target="_self" title="RLSUB_2:th.14">Th18</a></span>;<br/>
<a NAME="E4:43"/>
 <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">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="rlsub_2.html#D4" target="_self" title="RLSUB_2:def.4">Def4</a></span>;<br/>
<a NAME="E5:43"/><span class="kw">then </span>
 <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">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="c3">W2</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c2">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlsub_2.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:43"/>
<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="c3">W2</font>,<font color="Maroon" title="c2">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="rlsub_2.html#D4" target="_self" title="RLSUB_2:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
