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<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">W1</font>, <font color="Maroon" title="c3">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>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font>, <font color="Maroon" title="c7">u1</font>, <font color="Maroon" title="c8">u2</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>assume </b><a NAME="E1:63"/><i><font color="Green" title="E49">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="c2">W1</font>,<font color="Maroon" title="c3">W2</font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:63"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">v2</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u2</font> )
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green" title="E51">A3</font></i>: 
( <font color="Maroon" title="c5">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c7">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">W1</font> )
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c6">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font> &amp; <font color="Maroon" title="c8">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">C2</font> = <font color="Maroon" title="c5">v1</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c3">W2</font> as    <a href="rusub_1.html#M2" title="RUSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c3">W2</font> <b>by </b><i><a class="ref" href="rusub_1.html#D5" title="RUSUB_1:def.5">RUSUB_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">C1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font> as    <a href="rusub_1.html#M2" title="RUSUB_1:mode.2">Coset</a> of <font color="Maroon" title="c2">W1</font> <b>by </b><i><a class="ref" href="rusub_1.html#T68" title="RUSUB_1:th.68">RUSUB_1:68</a></i>;<br/>
<i><font color="Green" title="E53">A5</font></i>: <font color="Maroon" title="c7">u1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">u2</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E2:63"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>
<br/>.= 
<font color="Maroon" title="c5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">u2</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/>
<a NAME="E8:63"/>
<font color="Maroon" title="c6">v2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_2.html#E4:63"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="rusub_1.html#T17" title="RUSUB_1:th.17">RUSUB_1:17</a></i>;<br/>
<a NAME="E9:63"/><b>then </b>
( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C1</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C2</font> &amp; <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C1</font> &amp; <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C2</font> )
 
<b>by </b><i><a class="txt" href="rusub_2.html#E3:63"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="rusub_2.html#E7:63"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="rusub_2.html#E62"><i><font color="Green" title="E44">Lm16</font></i></a>, <a class="ref" href="rusub_1.html#T37" title="RUSUB_1:th.37">RUSUB_1:37</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E10:63"/><b>then </b><i><font color="Green" title="E54">A6</font></i>: 
( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">C2</font> &amp; <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">C2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">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="E12:63"/><i><font color="Green" title="E55">A7</font></i>: 
<font color="Maroon" title="c10">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="rusub_2.html#E1:63"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="rusub_2.html#T43" target="_self" title="RUSUB_2:th.43">Th43</a></i>;<br/>
<a NAME="E13:63"/><i><font color="Green" title="E56">A8</font></i>: 
( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> &amp; <font color="Maroon" title="c7">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> )
 
<b>by </b><i><a class="txt" href="rusub_2.html#E10:63"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="rusub_2.html#E12:63"><i><font color="Green" title="E55">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:63"/>
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font>
 ;<br/>

<b>thus </b><a NAME="E15:63"/>
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E2:63"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="rusub_2.html#E13:63"><i><font color="Green" title="E56">A8</font></i></a>, <a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/>


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