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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c5">y</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">X</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>











<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> &amp; not <font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">X</font> )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c6">W</font>, <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp10.html#T15" target="_self" title="VECTSP10:th.15">Th15</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">w1</font>, <font color="Maroon" title="c8">w2</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c4">X</font>;<br/><b>let </b><font color="Maroon" title="c11">r1</font>, <font color="Maroon" title="c12">r2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>;<br/>





<b>assume </b><a NAME="E3:31"/><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c7">w1</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c6">W</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c8">w2</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c6">W</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">x2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">z1</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c14">z2</font> = <font color="Maroon" title="c10">x2</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c16">y2</font> = <font color="Maroon" title="c10">x2</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="vectsp10.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c11">r1</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c12">r2</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r2</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font> )
 
<b>by </b><i><a class="txt" href="vectsp10.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/>
<a NAME="E7:31"/><b>then </b>
( <font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c16">y2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c6">W</font> )
 
<b>by </b><i><a class="txt" href="vectsp10.html#E2:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="vectsp10.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="vectsp10.html#T8" target="_self" title="VECTSP10:th.8">Th8</a></i>;<br/>
<a NAME="E8:31"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">y2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a></i>;<br/>
<a NAME="E9:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">r2</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">r2</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/>
<a NAME="E10:31"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">r2</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp10.html#T4" target="_self" title="VECTSP10:th.4">Th4</a></i>;<br/>
<a NAME="E11:31"/>
( <font color="Maroon" title="c7">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <font color="Maroon" title="c8">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="vectsp10.html#E2:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="vectsp10.html#E3:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="vectsp10.html#E6:31"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="vectsp10.html#T7" target="_self" title="VECTSP10:th.7">Th7</a></i>;<br/>
<b>then </b><i><font color="Green" title="E25">A7</font></i>: <font color="Maroon" title="c7">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">w2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">y2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">y2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">y2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">y2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">r2</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/>
<a NAME="E13:31"/><i><font color="Green" title="E26">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">r2</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">r2</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/>
<font color="Maroon" title="c15">y1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">y2</font> = 
<font color="Maroon" title="c13">z1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">z2</font>
<b>by </b><i><a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">x2</font>
<b>by </b><i><a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></i>
;<br/>
<b>hence </b><a NAME="E15:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">w2</font></span>)</span> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> <font color="Maroon" title="c6">W</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">r2</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="vectsp10.html#E2:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="vectsp10.html#E8:31"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="vectsp10.html#E10:31"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="vectsp10.html#E12:31"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="vectsp10.html#E13:31"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="vectsp10.html#T6" target="_self" title="VECTSP10:th.6">Th6</a></i>;<br/>


</div>
